I am working on a new problem that I gave myself. It is expanding on something I learned right here from one of you. I am not a mathematician nor did I major in it, but I wanted to see if what I did here is correct. Please help me with this as I have been thinking about it for some time now.
Problem: We found that it takes $3$ minutes per session to process each of the transactions listed below with the following number of reps. Our goal is to reach $0.5$ minutes by finding how many more reps are needed? Please see the figure below:
Question $1a:$ According to the chart, at $8:00$ am, there were $4$ reps who processed $32$ transactions in $3$ minutes ($180$ seconds). How many more reps do we need to minimize that time down to $0.5$ minutes?
Solution: First, we should throw out the number of transactions because it stays the same so it shouldn't matter. We would do $(4)(3)/0.5= 24$ reps total or $20$ more reps needed since we already have $4$ reps.
Question $1b:$ According to the chart, at $10: 00$ am, there were $9$ reps who processed $68$ transactions in $3$ minutes ($180$ seconds). How many more reps do we need to minimize that time down to $0.5$ minutes?
Solution: Same as previous problem. We do $(9)(3)/0.5=54$ reps total or $45$ more reps needed since we already have $9$ reps.
Question $2:$ What if we assumed according to the chart it took a total time of 3 minutes per session from Monday $-$ Thursday at $8:00$ am to process these total transactions with the total number of reps combined. We now want to attain $0.5$ minutes for those days of Monday $-$ Thursday at $8:00$ am. How many more reps would we have needed for those days total?
Solution: I'm not sure about this one but, I'll try. At $8:00$ am, there were a total of $70$ transactions from Monday $-$ Thursday. There were a total of $13$ reps from those days. According to the previous problem we did, the following solution should remain the same.
We would do $(13)(3)/0.5=78$ reps total or $65$ more reps needed since we already have $13$ reps.
Question $3a$: According to the chart, at $8:00$ am on Monday, there were $32$ transactions processed within $3$ minutes ($180$ seconds). How long would it take to process $16$ transactions?
Solution: We would do $\frac{32}{16}$ $=$ $\frac{3}{x}$. This would give us $1.5$ minutes.
Question $3b$: According to the chart, at $8:00$ am on Monday, there were $32$ transactions processed within $3$ minutes ($180$). How many transactions would be processed within $1.5$ minutes?
Solution: This is a reversal of question $3a$.
We would do $\frac{32}{x}$ $=$ $\frac{3}{1.5}$. This would give us $16$ transactions.
To be clear on the questions that are being asked, the chart just shows an outline. The questions itself is what needs to be solved for example in question $1a$ it states there are $4$ reps who process $32$ transactions in $3$ minutes. How many more reps are needed to break down that time to $0.5$ minutes? The solution is $24$ or $20$ more reps considering we already have $4$.
Can someone please review my solutions and tell me if they are correct? I put a lot of time and effort into this and would appreciate it if someone did.
If there is an easier solution of how to solve this it would be great if you can show me. Please show your working to your own solutions.